Richard Anderson Lecture 26 NP-Completeness CSE 421 Algorithms P Richard Anderson Lecture 26 NP-Completeness

NP Completeness: The story so far Circuit Satisfiability is NP-Complete

Background P: Class of problems that can be solved in polynomial time NP: Class of problems that can be solved in non-deterministic polynomial time Y is Polynomial Time Reducible to X Solve problem Y with a polynomial number of computation steps and a polynomial number of calls to a black box that solves X Notation: Y <P X Suppose Y <P X. If X can be solved in polynomial time, then Y can be solved in polynomial time A problem X is NP-complete if X is in NP For every Y in NP, Y <P X If X is NP-Complete, Z is in NP and X <P Z Then Z is NP-Complete

Today There are a whole bunch of other important problems which are NP-Complete

Populating the NP-Completeness Universe Circuit Sat <P 3-SAT 3-SAT <P Independent Set 3-SAT <P Vertex Cover Independent Set <P Clique 3-SAT <P Hamiltonian Circuit Hamiltonian Circuit <P Traveling Salesman 3-SAT <P Integer Linear Programming 3-SAT <P Graph Coloring 3-SAT <P Subset Sum Subset Sum <P Scheduling with Release times and deadlines NP P

Satisfiability Literal: A Boolean variable or its negation. Clause: A disjunction of literals. Conjunctive normal form: A propositional formula  that is the conjunction of clauses. SAT: Given CNF formula , does it have a satisfying truth assignment? 3-SAT: SAT where each clause contains exactly 3 literals. Any logical formula can be expressed in CNF. Ex: Yes: x1 = true, x2 = true x3 = false.

3-SAT is NP-Complete    Theorem. 3-SAT is NP-complete. Pf. Suffices to show that CIRCUIT-SAT  P 3-SAT since 3-SAT is in NP. Let K be any circuit. Create a 3-SAT variable xi for each circuit element i. Make circuit compute correct values at each node: x2 =  x3  add 2 clauses: x1 = x4  x5  add 3 clauses: x0 = x1  x2  add 3 clauses: Hard-coded input values and output value. x5 = 0  add 1 clause: x0 = 1  add 1 clause: Final step: turn clauses of length < 3 into clauses of length exactly 3. ▪ output x0  x1 x2   x5 x4 x3 ? ?

Independent Set Independent Set Graph G = (V, E), a subset S of the vertices is independent if there are no edges between vertices in S 1 2 3 5 4 6 7

3 Satisfiability Reduces to Independent Set Claim. 3-SAT  P INDEPENDENT-SET. Pf. Given an instance  of 3-SAT, we construct an instance (G, k) of INDEPENDENT-SET that has an independent set of size k iff  is satisfiable. Construction. G contains 3 vertices for each clause, one for each literal. Connect 3 literals in a clause in a triangle. Connect literal to each of its negations. G k = 3

3 Satisfiability Reduces to Independent Set Claim. G contains independent set of size k = || iff  is satisfiable. Pf.  Let S be independent set of size k. S must contain exactly one vertex in each triangle. Set these literals to true. Truth assignment is consistent and all clauses are satisfied. Pf  Given satisfying assignment, select one true literal from each triangle. This is an independent set of size k. ▪ and any other variables in a consistent way G k = 3

Vertex Cover Vertex Cover Graph G = (V, E), a subset S of the vertices is a vertex cover if every edge in E has at least one endpoint in S 1 2 3 5 4 6 7

IS <P VC Lemma: A set S is independent iff V-S is a vertex cover To reduce IS to VC, we show that we can determine if a graph has an independent set of size K by testing for a Vertex cover of size n - K

IS <P VC Find a maximum independent set S Show that V-S is a vertex cover 1 2 1 2 3 5 3 5 4 4 6 6 7 7

Clique Clique Graph G = (V, E), a subset S of the vertices is a clique if there is an edge between every pair of vertices in S 1 2 3 4 5 6 7

Complement of a Graph Defn: G’=(V,E’) is the complement of G=(V,E) if (u,v) is in E’ iff (u,v) is not in E 1 2 1 2 3 5 3 5 4 4 6 6 7 7

IS <P Clique Lemma: S is Independent in G iff S is a Clique in the complement of G To reduce IS to Clique, we compute the complement of the graph. The complement has a clique of size K iff the original graph has an independent set of size K

Hamiltonian Circuit Problem Hamiltonian Circuit – a simple cycle including all the vertices of the graph

Thm: Hamiltonian Circuit is NP Complete Reduction from 3-SAT

Traveling Salesman Problem Minimum cost tour highlighted Traveling Salesman Problem Given a complete graph with edge weights, determine the shortest tour that includes all of the vertices (visit each vertex exactly once, and get back to the starting point) 3 7 7 2 2 5 4 1 1 4 Find the minimum cost tour

Thm: HC <P TSP

Graph Coloring NP-Complete Polynomial Graph K-coloring